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From: mcohen@charming.nrtc.northrop.com (Martin Cohen)
Subject: Re: Is Turing Wrong about the Limits of Computation?
Message-ID: <D41xMB.496@gremlin.nrtc.northrop.com>
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Organization: Northrop Grumman Automation Sciences Laboratory, Pico Rivera, CA
References: <3h53g7$e18@highway.LeidenUniv.nl> <Pine.SUN.3.91.950207154346.8460A-100000@sun1> <3hcpei$85k@highway.LeidenUniv.nl>
Date: Wed, 15 Feb 1995 17:22:58 GMT
Lines: 42

In article <3hcpei$85k@highway.LeidenUniv.nl> vosse@ruls41.LeidenUniv.nl (Theo Vosse) writes:
>Eugen Leitl (ui22204@sunmail.lrz-muenchen.de) wrote:
............
>If the
>TM has got Q states, you get a total of at least QM(M+1)/2 possible
>configurations that the second TM has to remember in order to
>detect the cycle. But then that TM would have to have a tape
>larger than that of the original TM (i.e., it has got more memory),
>so you actually have a TM being controlled by one that is more
>powerful! This is not very acceptable (or would you get yourself
>a Cray to check the parity on your old XT's memory banks?).
>...................

There is an ingenious method of checking to see if a system is
in a loop (not original by me by any means):

Run two copies of the system, with the second running twice
as fast as the first. At each step of the first step,
check to see if the states of the two copies are the same.
If so, the machines are in a loop, and the period of the
loop divides the number of steps taken by the slower machine.

Proof: After n steps, machine 1 is in state M(n) and machine
2 is in state M(2n). If M(n)=M(2n) then we will have
M(2n)=M(3n)=...=M(kn) for all integral k>=1. If p is the
actual smallest period of M, then, if n=ap+b, 0<=b<p,
M(n)=M(n+ap)=M(2n)=M(n+ap+b), a contradiction unless b=0,
so p divides n evenly.

We thus trade time (have to run up to twice as long, but
usually not too much longer) and resources (have to run
two copies) for lots of space (to store all the states)
and lots of time (have to compare the current state
with all the previous states).

In general, this technique is a big win (IMHO).


-- 
Marty Cohen (mcohen@nrtc.northrop.com) - Not the guy in Philly
  This is my opinion and is probably not Northrop Grumman's!
          Use this material of your own free will
